Thermal Concentration and Poisson-Dirichlet Edge Statistics for Random-Lattice Gibbs Ensembles

Abstract

We study Gibbs measures on high-dimensional Haar-random unimodular lattices, where the energy of a lattice vector is its squared Euclidean norm. The random lattice is viewed as quenched geometric disorder, and $c>0$ denotes the scaled inverse temperature. We first analyze the edge window of vectors whose length is within the factor $e^{a/n}$ of the shortest length, with fixed $a$ as $n\to \infty$. For the full sign-class Gibbs ensemble, we prove a Poisson point process limit theorem for the Gibbs mass of this window. The mass vanishes in probability for $0<c\le 1$, while for $c>1$ it has a nontrivial Poisson limit, and the ranked Gibbs weights converge to the Poisson-Dirichlet distribution with parameter $1/c$. We then pass to a primitive-direction Gibbs ensemble and consider a fixed approximation factor ${\hat{I}}^3>1$. For this modified ensemble, we prove a weighted moment formula and a quenched thermal concentration result in the high-temperature range $0<c<1$. This yields the primitive fixed-factor visibility curve $c=γ^{-2}$ for approximate shortest directions. More precisely, the primitive Gibbs mass of the fixed-factor window tends to zero for $c<γ^{-2}$, to one for $γ^{-2}<c<1$, and to $1/2$ at the critical boundary $c=γ^{-2}$. Thus the fixed-factor theorem is a visibility statement for an idealized primitive target measure, not for the original full lattice Gibbs measure. The results provide a random-lattice thermodynamic reference model for Gibbs targets related to approximate shortest vectors, without implying an efficient algorithm for the shortest vector problem.

27 pages, no figure